An essentially algebraic glance to Kripke semantics: the S5 case

TL;DR

This paper explores the algebraic structure of finite S5-algebras, revealing their classification via Kan extensions of faithful symmetric group actions, thus connecting modal logic semantics with algebraic and categorical frameworks.

Contribution

It introduces a novel algebraic perspective on S5 Kripke semantics by linking finite S5-algebras to Kan extensions of symmetric group actions.

Findings

Finite S5-algebras classified by Kan extensions

Connection between algebraic models and symmetric group actions

Enhanced understanding of modal logic semantics

Abstract

We show that the category of finite $\textit{S5}$-algebras (dual to finite reflexive, symmetric and transitive Kripke frames) classifies the essentially algebraic theory whose models are Kan extensions of faithful actions of the finite symmetric groups.